Theorem al0ssb 5270

Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
al0ssb	⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Distinct variable group:   𝑦,𝑋

Proof of Theorem al0ssb

Step	Hyp	Ref	Expression
1		0ex 5269	. . 3 ⊢ ∅ ∈ V
2		sseq2 3973	. . . 4 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ ∅))
3		ss0b 4362	. . . 4 ⊢ (𝑋 ⊆ ∅ ↔ 𝑋 = ∅)
4	2, 3	bitrdi 286	. . 3 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅))
5	1, 4	spcv 3565	. 2 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 → 𝑋 = ∅)
6		0ss 4361	. . . 4 ⊢ ∅ ⊆ 𝑦
7	6	ax-gen 1797	. . 3 ⊢ ∀𝑦∅ ⊆ 𝑦
8		sseq1 3972	. . . 4 ⊢ (𝑋 = ∅ → (𝑋 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
9	8	albidv 1923	. . 3 ⊢ (𝑋 = ∅ → (∀𝑦 𝑋 ⊆ 𝑦 ↔ ∀𝑦∅ ⊆ 𝑦))
10	7, 9	mpbiri 257	. 2 ⊢ (𝑋 = ∅ → ∀𝑦 𝑋 ⊆ 𝑦)
11	5, 10	impbii 208	1 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Syntax hints:   ↔ wb 205  ∀wal 1539   = wceq 1541   ⊆ wss 3913  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-nul 5268

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4288

This theorem is referenced by:  iota0def  45392  aiota0def  45448